Question

If (log₂ (A + B) + log₂ (A + B))/2 = 1 + log₂ (7) and log₇ A – log₇ B = log₇ 5 – log₇ 2. Find the value of A² – B²
1. 44
2. 84
3. 90
4. 100

Answer 1

Correct Answer - Option 2 : 84

Calculation:

(log₂ (A + B) + log₂ (A + B))/2 = 1 + log₂ (7)

⇒ log₂ ((A + B) × (A + B))/2 = log₂ (2) + log₂ (7)

⇒ log₂ (A + B) ² = 2 × log₂ (2 × 7)

⇒ log₂ (A + B) ² = log₂ (14²)

⇒ (A + B) ² = 14²

⇒ (A + B) ² = 196      ….(1)

log₇ (A) – log₇ (B) = log₇ (5) – log₇ (2)

⇒ log₇ (A/ B) = log₇ (5/ 2)

⇒ A/ B = 5 / 2

⇒ A = 5 × B/ 2      ….(2)

Solving Equation (1) using Equation (2)

⇒ ((5 × B/2) + B) ² = 196

⇒ ((5 × B/2) + B) = 14

⇒ 7 B = 28

⇒ B = 4 and A = 10

⇒ A² – B² = 100 – 16

∴ Required answer is 84

x log (y) = log (y^x)

 log (a) - log (b) = log (a/b)        (This condition holds true only when bases of log under consideration are same)